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Meep unit transformation Wiki
This wiki should help to understand the transformation of physical quantities calculated by meep to other unit systems. The Meep unit system Because Meep is mainly used in photonics the majority of its users do not need the quantities calculated by meep in a standard units system like the International System of Units (SI). Hence, the output quantities of meep are given dimensionless. Although, users working in other fields of electrodynamics might be interested in transforming these quantities to different unit systems. This issue shall be addressed here. To access the unit transformation we start we the formulation of Maxwell's equations in meep. This specific formulation is the same as if the equations are formulated in the SI system. Hence, the unit transformation can be obtained by starting with the SI base units: There are seven basic units assigned to seven basic physical quantities which which all other physical quantities can be expressed. The basic physical quantities are sometimes also called as the "dimensions" of a quantity. For instance the velocity has the dimensions length/time. For the quantities in Electrodynamics, only four of these basic quantities are relevant: length l, time t, current I, and mass m. In the SI system the corresp458onding base units are m, s, A, and kg, respectively. But this is obviously an arbitrary choice. One can also choose different "units" to the measure the basic physical quantities, for instance a characteristic length scale of the observed system. This is the basic idea of the meep unit system: The physical quantities in meep are given dimensionsless, per definition. This can be done because Maxwell's equations have the inherent property of being scale invariant. So one can calculate the solution to Maxwell's equations with for the length 1 and rescale the solution afterwards with his/her choice of a unit length. The choice of the unit length and additional assumptions also determines the relation of the other basic physical quantities (see below). Given all the units these basic quantities, one can easily calculate any other quantity with them. The basic quantities Length: First, we start up the length scale. This can be chosen arbitrarily, so lets call it 'a'. 'a' stands for any unit scale of your choice. So it can me meters if you want to use the SI system but also micrometers, if you would prefore to observe microwaves and have convenient numbers. So we define: l:=a (1) meaning that we measure length in units of a (so a=1 in meep units). Time: For t we go a slightly different way. Here we define the time unit such that the vacuum velocity of light c=1 in meep. To obtain this, we take a look at the dimensions of c. Since c is a velocity they are l/t. So we choose t:=a/c, (2) leading to v=l/t=a/a*c=c. ''Example: ''If you observe a electromagnetic wave in meep propagating with the vacuum speed of light, its velocity in meep units v_m=1. So if you want know the velocity in your own units v_o you have to rescale it by performing v_o=c*v_m. But certainly you have to express c in your own units! For instance, in SI units you will observe: v_o=299792458*1 m/s. @Mischa: I want to put the meep 'c' example exactly here. Current and Mass: As we have chosen c=1 in meep units and we do know that c^2=1/(mu_0*eps_0), we find mu_0*eps_0=1. The most convenient way to fullfill this condition is to set mu_0:=1 and eps_0:=1. This definition can be used to derive the units of another quantity or a relation of two. Here, we aim for the latter and use the dimesions of permittivity t^4*I^2*m^-1*l^-3. With that and the previous assumption (eps_0=1 in meep units) we find: I^2/m=eps_0*c^4/a (3) Obviously, this only relates I and m and so we can define another basic unit arbitrarily. Here we define I:=I_0. (4) Instead of I you could use also m, or another quantity like for instance power. (For the latter you would have to express I oder m in (3) and in terms of power). With (4) we directly get: m=a*I_0^2/(eps_0*c^4). Category:Browse